Phylogenetics is a field primarily concerned with the reconstruction of the evolutionary history of present day species. Evolutionary history is often modeled by a phylogenetic tree, similar to a family tree. To recreate a phylogenetic tree from information about current species, one needs to make assumptions about the evolutionary process. These assumptions can range from full parametrised models of evolution to simple observations. This thesis looks at the reconstruction of phylogenetic trees under two different assumptions. The first, known as the ordinal assumption, has been previously studied and asserts that as species evolve, they become more dissimilar. The second, the convex assumption, has not previously been studied in this context and asserts that changes species go through to become dissimilar are progressively larger than the current differences between those species.

This thesis presents an overview of mathematical results in tree reconstruction from dissimilarity maps (also known as distance matrices) and develops techniques for reasoning about the ordinal and convex assumptions. In particular, three main results are presented: a complete classification of phylogenetic trees with four leaves under the ordinal assumption; a partial classification of phylogenetic trees with four leaves under the convex assumption; and, an independent proof of a result on the relationship between ultrametrics and the ordinal assumption.